LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite subfamilies of Nisnevich covers on the finite-type site

PDF p. 5: Finite refinements and integer sections

Statement

In the specified category of separated finite-type schemes over a field $k$ , every Nisnevich covering family ${U_{i} \to X}_{i \in I}$ contains a finite subfamily which is still Nisnevich. The empty family suffices if $X = \emptyset$ . This does not assert that the residue-field-lifting locus of a given étale map is Zariski open.

Facts & Assumptions

Given: The site above and the pointwise residue-field-lifting definition of a Nisnevich family.

Goal: Obtain a finite subfamily with that same pointwise lifting property.

[F1]

The completed interfaces specify these schemes and étale covering morphisms. Finite-type schemes over a field are Noetherian of finite dimension; closed reduced subschemes have finitely many irreducible components. Completely decomposed proper covers and the cdh topology, Nisnevich covers and distinguished squares

[F2]

On an integral affine scheme $Z = Spec A$ , a morphism from its generic point into a finite-type $Z$ -scheme extends over a nonempty open: choose an affine target neighbourhood over $Z$ , write its coordinate ring using finitely many $A$ -algebra generators and relations, and clear the finitely many denominators of their images in $Frac A$ . The resulting homomorphism into $A [1 / a]$ is the required extension. Relations hold there because $A [1 / a] ↪ Frac A$ is injective. This argument proves the fact rather than importing a rational-section theorem.

Proof

1.1F1F2given

For a nonempty reduced closed subscheme $Z \subset X$ and each of its finitely many generic points $η$ , choose a residue-preserving lift in one of the $U_{i}$ . This gives a generic section of $U_{i} \times_{X} Z_{η} \to Z_{η}$ , where $Z_{η}$ is the corresponding integral component. By [F2] it extends over a dense open of that component. Shrink this open to avoid the other components. It is then open also in $Z$ . Finitely many such opens have union $O \subset Z$ containing all generic points of $Z$ ; the finitely many chosen maps provide residue-preserving lifts over $O$ . A section preserves residue fields because the induced field maps $κ (z) \to κ (u) \to κ (z)$ have composite identity, so the middle field is the base field.

2.1F1step 1.1∎

The closed reduced complement $Z ∖ O$ has strictly smaller dimension than $Z$ . Repeat step 1.1 there. Induction on the finite dimension terminates, and at each stage only finitely many original indices are selected. Thus their union is finite and provides residue-preserving lifts at every point of $X$ . Each selected original map remains étale, so it is a Nisnevich subfamily. Nilpotents do not change points or residue fields; starting with $X_{r e d}$ therefore proves the claim for $X$ as well. No lifting-locus openness has been asserted.

Depends on

Used by · 2

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.